1) Research about the startup and basically ask them about their biggest obvious problems.
If they answer in shallow standard (aka "💩 i tell VCs") answers that's a red flag.
If they are honest and tell you their guesses it's a good sign
You won't be able to solve problems if the team doesn't address them honestly.
2) Ask the person why they work at the company (as warm up), what they would change if they could change anything and then if they are happy at this job.
This usually surprises people and you can tell a lot about the company with their initial reactions to those questions

Learn about the problem that the startup is solving. As @andreasklinger mentioned, research the company well and figure out if their answers are honest or just the usual VC pitch.
Very important to understand where you and your role fits into the overall company strategy. Given that most startup teams are small, it is easy to get sucked into a lot of 'other' stuff. While it may be okay to take on more work, you must ensure that you are adding value to the core business that the company is in.
Again, since startup teams are small, it is important to ask questions around the company culture. Will you face 'teething' problems when you join or will the role and responsibilities be well defined?
Ask them about their roadmap. What is their strategy to beat competition and grow.
While one learns a lot working at a startup, do ask about remuneration and appraisals. Given that you will be joining from Facebook, you may be used to certain perks :-)

@demiliag The motive of that question is find whether the founders are passionate about their startup.
There are startups that they build products to interest investors and/or unicorns to get acquired. You should try to avoid this pitfall. I think you get the picture here :)